Seminars and Colloquia by Series

Erdos-Posa theorems for undirected group-labelled graphs

Series
Dissertation Defense
Time
Friday, June 10, 2022 - 11:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006 (hybrid)
Speaker
Youngho YooGeorgia Tech

Erdos and Posa proved in 1965 that cycles satisfy an approximate packing-covering duality. Finding analogous approximate dualities for other families of graphs has since become a highly active area of research due in part to its algorithmic applications. In this thesis we investigate the Erdos-Posa property of various families of constrained cycles and paths by developing new structural tools for undirected group-labelled graphs.

Our first result is a refinement of the flat wall theorem of Robertson and Seymour to undirected group-labelled graphs. This structure theorem is then used to prove the Erdos-Posa property of A-paths of length 0 modulo p for a fixed odd prime p, answering a question of Bruhn and Ulmer. Further, we obtain a characterization of the abelian groups G and elements g for which A-paths of weight g satisfy the Erdos-Posa property. These results are from joint work with Robin Thomas.

We extend our structural tools to graphs labelled by multiple abelian groups and consider the Erdos-Posa property of cycles whose weights avoid a fixed finite subset in each group. We find three types of topological obstructions and show that they are the only obstructions to the Erdos-Posa property of such cycles. This is a far-reaching generalization of a theorem of Reed that Escher walls are the only obstructions to the Erdos-Posa property of odd cycles. Consequently, we obtain a characterization of the sets of allowable weights in this setting for which the Erdos-Posa property holds for such cycles, unifying a large number of results in this area into a general framework. As a special case, we characterize the integer pairs (L,M) for which cycles of length L mod M satisfy the Erdos-Posa property. This resolves a question of Dejter and Neumann-Lara from 1987. Further, our description of the obstructions allows us to obtain an analogous characterization of the Erdos-Posa property of cycles in graphs embeddable on a fixed compact orientable surface. This is joint work with Pascal Gollin, Kevin Hendrey, O-joung Kwon, and Sang-il Oum.

Zoom link: https://gatech.zoom.us/j/96860495360?pwd=cktMRVVqMDRtVnJsb3ZLRll1bFRJQT09

Learning Dynamics from Data Using Optimal Transport Techniques and Applications

Series
Dissertation Defense
Time
Wednesday, June 1, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Shaojun Ma

Zoom link: https://gatech.zoom.us/j/4561289292

Abstract: In recent years we have seen the popularity of optimal transport and deep learning. Optimal transport theory works well in studying differences among distributions, while deep learning is powerful to analyze high dimensional data. In this presentation we will discuss some of our recent work that combine both optimal transport and deep learning on data-driven problems. We will cover four parts in this presentation. The first part is studying stochastic behavior from aggregate data where we recover the drift term in an SDE, via the weak form of Fokker-Planck equation. The second part is applying Wasserstein distance on the optimal density control problem where we parametrize the control strategy by a neural network. In the third part we will show a novel form of computing Wasserstein distance, geometric and map all together in a scalable way. And in the final part, we consider an inverse OT problem where we recover cost function when an observed policy is given.

Contact geometric theory of Anosov flows in dimension three

Series
Dissertation Defense
Time
Wednesday, May 25, 2022 - 11:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Surena HozooriGeorgia Institute of Technology

Zoom link : https://gatech.zoom.us/j/98171168149

Since their introduction in the early 1960s, Anosov flows have defined an important class of dynamics, thanks to their many interesting chaotic features and rigidity properties. Moreover, their topological aspects have been deeply explored, in particular in low dimensions, thanks to the use of foliation theory in their study. Although the connection of Anosov flows to contact and symplectic geometry was noted in the mid 1990s by Mitsumatsu and Eliashberg-Thurston, such interplay has been left mostly unexplored. I will present some recent results on the contact and symplectic geometric aspects of Anosov flows in dimension 3, including in the presence of an invariant volume form, which is known to have grave consequences for the dynamics of these flows. Time permitting, the interplay of Anosov flows with Reeb dynamics, Liouville geometry and surgery theory will be briefly discussed as well.

Symmetric Tropical Rank 2 Matrix Completion

Series
Other Talks
Time
Monday, May 23, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
May Cai

An important recent topic is matrix completion, which is trying to recover a matrix from a small set of observed entries, subject to particular requirements. In this talk, we discuss results on symmetric tropical and symmetric Kapranov rank 2 matrices, and establish a technique of examining the phylogenetic tree structure obtained from the tropical convex hulls of their columns to construct the algebraic matroid of symmetric tropical rank 2 $n \times n$ matrices. This matroid directly answers the question of what entries of a symmetric $n \times n$ matrix needs to be specified generically to be completable to a symmetric tropical rank 2 matrix, as well as to a symmetric classical rank 2 matrix.

This is based on joint work with Cvetelina Hill and Kisun Lee.

Concentration of the Chromatic Number of Random Graphs

Series
Graph Theory Seminar
Time
Tuesday, May 17, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Lutz WarnkeUCSD
What can we say about the chromatic number \chi(G_{n,p}) of an n-vertex binomial random graph G_{n,p}? From a combinatorial perspective, it is natural to ask about the typical value of \chi(G_{n,p}), i.e., upper and lower bounds that are close to each other. From a probabilistic combinatorics perspective, it is also natural to ask about the concentration of \chi(G_{n,p}), i.e., how much this random variable varies. Among these two fundamental questions, significantly less is known about the concentration question that we shall discuss in this talk. In terms of previous work, in the 1980s Shamir and Spencer proved that the chromatic number of the binomial random graph G_{n,p} is concentrated in an interval of length at most \omega\sqrt{n}, and in the 1990s Alon showed that an interval of length \omega\sqrt{n}/\log n suffices for constant edge-probabilities p\in (0,1). In this talk, we prove a similar logarithmic improvement of the Shamir-Spencer concentration results for the sparse case p=p(n) \to 0, and also discuss several intriguing questions about the chromatic number \chi(G_{n,p}) that remain open. Based on joint work with Erlang Surya; see https://arxiv.org/abs/2201.00906

Approximation of invariant manifolds for Parabolic PDEs over irregular domains

Series
CDSNS Colloquium
Time
Friday, May 13, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Online via Zoom
Speaker
Jorge GonzalezGeorgia Tech

Please Note: Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09

 

The computation of invariant manifolds for parabolic PDE is an important problem due to its many applications. One of the main difficulties is dealing with irregular high dimensional domains when the classical Fourier methods are not applicable, and it is necessary to employ more sophisticated numerical methods. This work combines the parameterization method based on an invariance equation for the invariant manifold, with the finite element method. We implement the method for a variety of example problems having both polynomial and non-polynomial nonlinearities, on non-convex two-dimensional polygonal domains (not necessary simply connected), for equilibrium solutions. We implement a-posteriori error indicators which provide numerical evidence of the accuracy of the computations. This is a joint work with J.D Mireles-James, and Necibe Tuncer.  

Thesis defense: Invariance of random matrix

Series
Time
Thursday, May 12, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
JunTao DuanGeorgia institute of technology

Zoom link: https://gatech.zoom.us/j/92161924238

Random matrix has been found useful in many real world applications. The celebrated Johnson-Lindenstrauss lemma states that certain geometric structure of deterministic vectors is preserved when projecting high dimensional space $R^n$ to a lower dimensional space $R^m$. However, when random vectors are concerned, it is still unclear how the distribution of the geometry is affected by random matrices. Since random projection or embedding introduces dependence to independent random vectors, does it imply random matrices are inferior for transforming random vectors?

We will start with establishing a new  central limit theorem  for random variables with certain product dependence structure. At the same time, we obtain its Berry-Esseen type rate of convergence. Then we apply this general central limit theorem to random projections and embeddings of two independent random vectors $X, Z$. In particular, we show the distribution of inner product structure is preserved by random matrices. Roughly speaking, two independent random vectors remain "independent" in the randomly projected lower dimensional space or randomly embedded high dimensional space. More importantly, we also quantitatively characterize the distortion of distribution introduced by random matrices. The error term has a bound at most $O(\frac{1}{\sqrt{m}} + \frac{1}{\sqrt{n}})$. 

Then we also establish the fact that random matrices have low distortion on the norm of a random vector. It is first justified by establishing concentration of the projected or embedded norm under sub-Gaussian assumptions. A central limit theorem for the randomly projected norm is established as well similar to the CLT for inner product.

An army of one: stable solitary states in the second-order Kuramoto model

Series
CDSNS Colloquium
Time
Friday, May 6, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005; streaming via Zoom available
Speaker
Igor BelykhGeorgia State University

Please Note: Link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09

Symmetries are  fundamental concepts in modern physics and biology. Spontaneous symmetry breaking often leads to fascinating  dynamical patterns such as  chimera states in which structurally and dynamically identical oscillators  split into coherent and incoherent clusters.  Solitary states in which one oscillator separates from the coherent cluster and oscillates with a different frequency represent  “weak” chimeras. While a rigorous stability analysis of a “strong” chimera with a multi-oscillator incoherent cluster  is typically out of reach for finite-size networks, solitary states offer a unique test bed for the development of stability approaches to large chimeras. In this talk, we will present such an approach and study the stability of solitary states in Kuramoto networks of identical 2D phase oscillators with inertia and a phase-lagged coupling.   We will derive asymptotic stability conditions for such solitary states as a function of inertia, network size, and phase lag that may yield either attractive or repulsive coupling. Counterintuitively, our analysis demonstrates that (i) increasing the size of the coherent cluster can promote the stability of the solitary state in the attractive coupling case and (ii) the solitary state can be stable in small-size networks with all repulsive coupling. We also discuss the implications of our analysis for the emergence of rotatory chimeras and splay states. This is a joint work with V. Munyaev, M. Bolotov, L. Smirnov, and G. Osipov.

 

Two conjectures on the spread of graphs

Series
Combinatorics Seminar
Time
Friday, April 29, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael TaitVillanova University

Given a graph $G$ let $\lambda_1$ and $\lambda_n$ be the maximum and minimum eigenvalues of its adjacency matrix and define the spread of $G$ to be $\lambda_1 - \lambda_n$. In this talk we discuss solutions to a pair of 20-year-old conjectures of Gregory, Hershkowitz, and Kirkland regarding the spread of graphs.
   
The first, referred to as the spread conjecture, states that over all graphs on $n$ vertices the join of a clique of order $\lfloor 2n/3 \rfloor$ and an independent set of order $\lceil n/3 \rceil$ is the unique graph with maximum spread. The second, referred to as the bipartite spread conjecture, says that for any fixed $e\leq n^2/4$, if $G$ has maximum spread over all $n$-vertex graphs with $e$ edges, then $G$ must be bipartite.

We show that the spread conjecture is true for all sufficiently large $n$, and we prove an asymptotic version of the bipartite spread conjecture. Furthermore, we exhibit an infinite family of counterexamples to the bipartite spread conjecture which shows that our asymptotic solution is tight up to a multiplicative factor in the error term. This is joint work with Jane Breen, Alex Riasanovsky, and John Urschel.

Back to boundaries in billiards

Series
CDSNS Colloquium
Time
Friday, April 29, 2022 - 13:00 for
Location
Zoom Link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09
Speaker
Yaofeng SuSoM, GT

Abstract: This talk has 4 or 5 parts

  1. I will start with a physical toy model to introduce billiards/open billiards, which describe the dynamics of a particle in a compact manifold/in a particular open area of this manifold.

  2. One of the main questions of open billiards is Poisson approximations. It describes the asymptotic behavior of the dynamics in statistical distributions.  I will define it for billiards systems.

  3. The main result is that such approximations hold for a billiard system that has arbitrarily slow chaos.

  4. I will sketch the idea of the proof.

  5. If time permits, I will talk about the connection between this work and riemann hypothesis.

This is a joint work with Prof. Leonid Bunimovich.

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